Infinite-Dimensionality modulo Absolute Borel Classes

Vitalij Chatyrko; Yasunao Hattori

Bulletin of the Polish Academy of Sciences. Mathematics (2008)

  • Volume: 56, Issue: 2, page 163-176
  • ISSN: 0239-7269

Abstract

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For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces X α , Y α and Z α such that (i) f X α , f Y α , f Z α = ω ₀ , where f is either trdef or ₀-trsur, (ii) A ( α ) - t r i n d X α = ∞ and M ( α ) - t r i n d X α = - 1 , (iii) A ( α ) - t r i n d Y α = - 1 and M ( α ) - t r i n d Y α = ∞ , and (iv) A ( α ) - t r i n d Z α = M ( α ) - t r i n d Z α = ∞ and A ( α + 1 ) ∩ M ( α + 1 ) - t r i n d Z α = - 1 . We also show that there exists no separable metrizable space W α with A ( α ) - t r i n d W α ≠ ∞ , M ( α ) - t r i n d W α ≠ ∞ and A ( α ) ∩ M ( α ) - t r i n d W α = ∞ , where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

How to cite

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Vitalij Chatyrko, and Yasunao Hattori. "Infinite-Dimensionality modulo Absolute Borel Classes." Bulletin of the Polish Academy of Sciences. Mathematics 56.2 (2008): 163-176. <http://eudml.org/doc/281221>.

@article{VitalijChatyrko2008,
abstract = {For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_α$, $Y_α$ and $Z_α$ such that (i) $fX_α, fY_α, fZ_α = ω₀$, where f is either trdef or ₀-trsur, (ii) $A(α)-trind X_α = ∞$ and $M(α)-trind X_α = -1$, (iii) $A(α)-trind Y_α = -1$ and $M(α)-trind Y_α = ∞$, and (iv) $A(α)-trind Z_α = M(α)-trind Z_α = ∞$ and $A(α+1) ∩ M(α+1)-trind Z_α = -1$. We also show that there exists no separable metrizable space $W_α$ with $A(α)-trind W_α ≠ ∞$, $M(α)-trind W_α ≠ ∞$ and $A(α) ∩ M(α)-trind W_α = ∞$, where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.},
author = {Vitalij Chatyrko, Yasunao Hattori},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {small transfinite inductive dimension modulo ; separable metrizable space; absolute Borel class},
language = {eng},
number = {2},
pages = {163-176},
title = {Infinite-Dimensionality modulo Absolute Borel Classes},
url = {http://eudml.org/doc/281221},
volume = {56},
year = {2008},
}

TY - JOUR
AU - Vitalij Chatyrko
AU - Yasunao Hattori
TI - Infinite-Dimensionality modulo Absolute Borel Classes
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2008
VL - 56
IS - 2
SP - 163
EP - 176
AB - For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_α$, $Y_α$ and $Z_α$ such that (i) $fX_α, fY_α, fZ_α = ω₀$, where f is either trdef or ₀-trsur, (ii) $A(α)-trind X_α = ∞$ and $M(α)-trind X_α = -1$, (iii) $A(α)-trind Y_α = -1$ and $M(α)-trind Y_α = ∞$, and (iv) $A(α)-trind Z_α = M(α)-trind Z_α = ∞$ and $A(α+1) ∩ M(α+1)-trind Z_α = -1$. We also show that there exists no separable metrizable space $W_α$ with $A(α)-trind W_α ≠ ∞$, $M(α)-trind W_α ≠ ∞$ and $A(α) ∩ M(α)-trind W_α = ∞$, where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.
LA - eng
KW - small transfinite inductive dimension modulo ; separable metrizable space; absolute Borel class
UR - http://eudml.org/doc/281221
ER -

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